14.15: Problems

Problems

1. When we express the energy of a system as a function of entropy, volume, and composition, we have \(E=E\left(S,V,n_1,n_2,\dots ,\ n_{\omega }\right)\). Since \(S\) and \(V\) are extensive variables, we have \(\lambda E=E\left(\lambda S,\lambda V,{\lambda n}_1,\lambda n_2,\dots ,\ \lambda n_{\omega }\right)\). Find \({\left({\partial \left(\lambda E\right)}/{\partial \lambda }\right)}_{SV}\). From this result, show that \[G=\sum^{\omega }_{j=1}{{\mu }_jn_j} \nonumber \]

2. When we express the energy of a system as a function of pressure, temperature, and composition, we have \(E=E\left(P,T,n_1,n_2,\dots ,\ n_{\omega }\right)\). Because P and T are independent of \(\lambda\), \(\lambda E=E\left(P,T,{\lambda n}_1,\lambda n_2,\dots ,\ \lambda n_{\omega }\right)\). Show that

\[E=\sum^{\omega }_{j=1} \overline{E}_jn_j \nonumber \]

3. From \(E\mathrm{=E}\left(P,T,n_{\mathrm{1}},n_{\mathrm{2}}\mathrm{,\dots ,\ }n_{\omega }\right)\) and the result in problem 2, show that

\[\left[{\left(\frac{\partial H}{\partial T}\right)}_P + P{\left(\frac{\partial V}{\partial T}\right)}_P\right]dT + \left[P{\left(\frac{\partial V}{\partial P}\right)}_T + T{\left(\frac{\partial V}{\partial T}\right)}_P\right]dP = \sum^{\omega }_{j\mathrm{=1}}{n_j}d{\overline{E}}_j \nonumber \]

Note that at constant pressure and temperature,

\[\sum^{\omega }_{j\mathrm{=1}}{n_j}d{\overline{E}}_j\mathrm{=0} \nonumber \]

4. If pressure and temperature are constant, \(E=E\left(n_1,n_2,\dots ,\ n_{\omega }\right)\) and \(\lambda E=E\left({\lambda n}_1,\lambda n_2,\dots ,\ \lambda n_{\omega }\right)\). Show that \(\sum^{\omega }_{j\mathrm{=1}}{n_j}d{\overline{E}}_j\mathrm{=0}\) follows from these relationships.

5. A solution contains \(n_1\) moles of component 1, \(n_2\) moles of component 2, \(n_3\) moles of component 3, etc. Let \(n=n_1+n_2+n_3+...\) The mole fraction of component \(j\) is \(x_j={n_j}/{n}\). Show that \[\left(\frac{\partial x_j}{\partial n_j}\right)=\frac{n-n_j}{n^2} \nonumber \] and, for \(j\neq k\), \[\ \left(\frac{\partial x_j}{\partial n_k}\right)=\frac{-n_j}{n^2} \nonumber \] What are \[\left(\frac{\partial x_1}{\partial n_1}\right) \nonumber \] and \[\left(\frac{\partial x_2}{\partial n_2}\right) \nonumber \] if the solution has only two components?

6. For any extensive state function, \(Y\left(P,T,n_1,n_2,\dots ,\ n_{\omega }\right)\), the arguments developed in this chapter lead, at constant \(P\) and\(\ T\), to the equations

\[Y=n_1{\overline{Y}}_1+n_2{\overline{Y}}_2+\dots +n_{\omega }{\overline{Y}}_{\omega } \nonumber \] and \[0=n_1d{\overline{Y}}_1+n_2d{\overline{Y}}_2+\dots +n_{\omega }d{\overline{Y}}_{\omega } \nonumber \]

Where \({\overline{Y}}_j\) is the partial molar quantity \({\left({\partial Y}/{\partial n_j}\right)}_{P,T,n_{m\neq j}}\).

(a) Prove that \(0=x_1d{\overline{Y}}_1+x_2d{\overline{Y}}_2+\dots +x_{\omega }d{\overline{Y}}_{\omega }\)

(b) Prove that \[0=n_1\left(\frac{\partial {\overline{Y}}_1}{\partial n_1}\right)+n_2\left(\frac{\partial {\overline{Y}}_2}{\partial n_2}\right)+\dots +n_{\omega }\left(\frac{\partial {\overline{Y}}_{\omega }}{\partial n_{\omega }}\right) \nonumber \] (c) Prove that \[0=x_1\left(\frac{\partial {\overline{Y}}_1}{\partial x_1}\right)+x_2\left(\frac{\partial {\overline{Y}}_2}{\partial x_2}\right)+\dots +x_{\omega }\left(\frac{\partial {\overline{Y}}_{\omega }}{\partial x_{\omega }}\right) \nonumber \]

7. The enthalpy of mixing is measured in a series of experiments in which solid solute, \(A\), dissolves to form an aqueous solution. These enthalpy data are represented well by empirical equations \({\Delta }_{mix}H={\alpha }_1\underline{m}+{\alpha }_2{\underline{m}}^2\), \({\alpha }_1={\beta }_{11}+{\beta }_{12}\left(T-273.15\right)\) and

\({\alpha }_2={\beta }_{21}+{\beta }_{22}\left(T-273.15\right)\) with \[{\beta }_{11}=10.0\ \mathrm{kJ}\ {\mathrm{molal}}^{-1} \nonumber \] \[{\beta }_{12}=-0.14\ \mathrm{kJ}\ {\mathrm{molal}}^{-2}\ K^{-1} \nonumber \] \[{\beta }_{21}=-3.00\ \mathrm{kJ}\ {\mathrm{molal}}^{-1} \nonumber \] \[{\beta }_{22}=-0.040\ \mathrm{kJ}\ {\mathrm{molal}}^{-2}\ K^{-1} \nonumber \] Find \({\overline{L}}_A\), \({\overline{L}}_{H_2O}\), \({\overline{J}}_A\), and \({\overline{J}}_{H_2O}\) as functions of \({\underline{m}}_A\) and \(T\). Find \({\overline{L}}_A\), \({\overline{L}}_{H_2O}\), \({\overline{J}}_A\), and \({\overline{J}}_{H_2O}\) for a one molal solution at 209 K. What is the value of

\[{ \ln \frac{{\tilde{a}}_A\left(1\mathrm{\ molal},290\mathrm{\ K}\right)}{{\tilde{a}}_A\left(1\mathrm{\ molal},273.15\mathrm{\ K}\right)}\ } \nonumber \]

Notes

\({}^{1}\) We can make other assumptions. It is possible to describe an inhomogeneous system as a collection of many macroscopic, approximately homogeneous regions.